Show that a wf $\mathscr{B}(t)⇒(\exists x_i)\mathscr{B}(x_i)$ logically valid if $t$ is free for $x_i$ in $\mathscr{B}$

Question

In Mendelson's Introduction to Mathematical Logic, exercise 2.18 asks us to show that certain wfs are logically valid. A wf B is said to be logically valid if and only if B is true for every interpretation.

Given the following wf:

Mendelson provides the following answer at the back of the book:

My question has to do with the first part of the answer. I understand how "By (X), $(∀x_i)\lnot \mathscr{B}(x_i)\rightarrow\lnot \mathscr{B}(t)$" is logically valid - this is because $\lnot\mathscr{B}(x_i)$ is also a wf from the definition of the language, right?

And, by (III), $\mathscr{B}(t)⇒\lnot(\forall x_i)\lnot\mathscr{B}(x_i)$ is logically valid - this is because as property (III) states, if $(∀x_i)\lnot \mathscr{B}(x_i)\rightarrow\lnot \mathscr{B}(t)$ is true for the interpretation $M$ and $[(\forall x_i)\lnot\mathscr{B}(x_i)\rightarrow\lnot\mathscr{B}(t)]\rightarrow[\mathscr{B}(t)\rightarrow\lnot(\forall x_i)¬\mathscr{B}(x_i)]$ then $\mathscr{B}(t)\rightarrow\lnot(\forall x_i)¬\mathscr{B}(x_i)$ is true for the interpretation $M$.

How though, does Mendelson's obtain $[(\forall x_i)\lnot\mathscr{B}(x_i)\rightarrow\lnot\mathscr{B}(t)]\rightarrow[\mathscr{B}(t)\rightarrow\lnot(\forall x_i)¬\mathscr{B}(x_i)]$ ? I understand that it is a tautology but I do not see how it is he comes to it?

If it is obvious that I've made a jump over a gap in knowledge any help would be greatly appreciated!

Answer 1

What Mendelson is doing is the following:

• Realize that $\exists x_i. B(x_i)$ is equivalent to $\lnot \forall x_i. \lnot B(x_i)$;
• Prove $B(t) \implies \lnot \forall x_i. \lnot B(x_i)$ by proving its contraposition $\forall x_i. \lnot B(x_i) \implies \lnot B(t)$.

This second step is, of course, using that particular tautology.